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Math Mechanics: Function Transformations

The Pattern

Function transformations are a fundamental part of Algebra II and Trigonometry – but few students actually understand them terribly well. They learn the formulas for lines, parabolas, square root functions, logarithmic functions, trigonometric functions, and conic sections as if they were all unique… when, in actuality, each of these formulas shares at least 80% of its contents with each other.

Let’s take a look at Point-Slope Form for linear functions first.

Here, we have the archetypical line graph: it passes through the point (h, k) with a slope of m. Effectively, it’s been translated right by h, up by k, and vertically stretched by m times. Not too bad!

Can we see any patterns with the formula for quadratic functions?

It’s almost identical! We’ve switched m out for A, but it still acts as the same thing: our vertical stretch factor. Once again, it’s been shifted to the right by h and up by k. This time, we’re especially interested in that point, as it’s the parabola’s vertex, but transforming it is just as easy.

Let’s take a few more at once this time:

All of these look almost identical! The only one that’s remotely different is the trigonometric graph, which introduces B – our horizontal dilation. There’s clearly a pattern here; let’s figure out what it is.

The Rules

Let’s take a look at that final function one more time. Its parent function is y = sin(x). Now, let’s take a look at what happens inside of those parentheses when we transform it.

To horizontally stretch the function by B times, we divided x by B.

If we think about it, this checks out: for instance, if B were 3, then when we had an x value of, say, 6, the sin function would be evaluating it as if it were 2, even though we had gone to the right three times as much as usual! This is easiest to see with linear functions: if you graph y = x/3, it’ll get three times wider than usual.

To horizontally shift the function by h, we subtracted h from x.

Again, this checks out: if we have x – 2, then when x = 2, we’ll be at the same y-coordinate as we would have when x = 0 for our original function. It’s a little tricky, but drawing a picture of it might help you make the connection!

(Special note: be very careful when you have both an x-shift and an x-dilation: it’s very easy to get confused as to how far you’re actually moving right or left!)

From here, we have our first important conclusion: everything that changed an element related to x happened within the parentheses.

Our translations related to y are usually quite intuitive: outside of the parentheses, we multiply by A to vertically stretch by that many times and add k to shift up by that amount.

Everything that changed an element related to y happened outside of the parentheses – and it worked the opposite way to how it worked for x. Instead of subtracting h, we added k. Instead of dividing by B, we multiplied by A. They’re opposites!

The “Rulebreakers”

We’ve got everything nicely tied up! We’ve found the pattern, everything’s nice and consistent, and we’re ready to go home and have a nice nap. At least, it looks that way… until we take a look at conic sections.

“Conic sections” is a phrase used to refer to parabolas, circles, ellipses, and hyperbolas – and they shatter what we thought we knew about x transformations and y transformations being opposites. Let’s take a look at the equation for a circle centered at (h, k).

Uh oh. Now both x and y have their translations subtracted from them. But surely circles are a one-off thing, right? Not a symptom of a larger misunderstanding? Let’s take a look at ellipses and hyperbolas; surely those will fix our problems.

That… didn’t help. But it did reveal one new thing: our good old friends A and B are back – and just like h and k, they’re also changing things up. In fact, everything about our y transformations is working just like our x transformations now. That gives me an idea; let’s take a look back at our trigonometric function and see if we can make some sense out of this.

Oho! It’s actually completely consistent, and has been this entire time! All the transformations that affected x are in the parentheses on the right, and all the transformations that affected y are in the parentheses on the left – and they’re both exactly the same as each other!

We can finally finish our rules; this time, they’re actually consistent and reliable. For ANY function, if you want to shift it right, subtract h from x. If you want to stretch it in the x-direction, divide x by B. If you want to vertically shift it, subtract k from y. If you want to stretch it in the y-direction, divide y by A.

(Extra credit: see if you can figure out why the equation for an ellipse looks that way. As a hint, consider it as a circle with a radius of 1, then figure out what A and B do to the function!)

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